SAURAV LUTHRA, MTH643 PROBLEM SET 2

Overview.

Placeholder - summary

HP Helmerich & Payne, Inc. (NYSE)

PDS Precision Drilling Corporation (NYSE)

QUESTION 1.

Past 5 Years of Prices and Log Returns

The plots of the HP and PDS stock price are both very similar since they are similar companies operating in the same industry. Although the price range over the last 5 years is approximately $10-$50 for HP and $10-$80 for PDS, their trends and moves are very similar. They both experienced high volatility at times of oil price shocks and market uncertainty. Particularly in 2020 (COVID-19 Pandemic) and 2022/23 (Ukraine Invasion). Neither of them have a clear mean value through time and have long-term changes in value. Their variances (volatility) are also not at all constant throughout time. It is difficult to say with perfect certainty the degree of stationarity in either of the series, but considering the fact that the company and stock evolves fundamentally over time, the stock mean price varies greatly depending on the time period/window you consider, and the variance/volatility also fluctuate greatly, I would not consider any of these series to be stationary.

# ==============================================================================
# SETUP AND GET DATA FOR HP

getSymbols("HP", src = "yahoo", from = as.Date("2020-02-01", to = as.Date("2025-02-01")))
## [1] "HP"
HP_prices <- (HP$HP.Adjusted)

HP_log_returns <- as.vector(diff(log(HP_prices))[-1])
HP_log_returns_sorted <- sort(HP_log_returns)

HP_n <- length(HP_log_returns_sorted)

# ==============================================================================
# SETUP AND GET DATA FOR PDS

getSymbols("PDS", src = "yahoo", from = as.Date("2020-02-01", to = as.Date("2025-02-01")))
## [1] "PDS"
PDS_prices <- (PDS$PDS.Adjusted)

PDS_log_returns <- as.vector(diff(log(PDS_prices))[-1])
PDS_log_returns_sorted <- sort(PDS_log_returns)

PDS_n <- length(PDS_log_returns_sorted)

# ==============================================================================
# PLOT PRICES FOR HP AND PDS

# plot HP price series
plot(index(HP_prices), HP_prices, type = "l", col = "orangered", lwd = 1, main = "Closing-Price of HP", xlab = "Date", ylab = "Price ($)")

# plot PDS price series
plot(index(PDS_prices), PDS_prices, type = "l", col = "mediumturquoise", lwd = 1, main = "Closing-Price of PDS", xlab = "Date", ylab = "Price ($)")

QUESTION 2.

Plots of Log-Returns

The plots of the HP and PDS log returns are both very similar since they are similar companies operating in the same industry. Over the last 5 years, HP and PDS both have log returns that are tightly banded slightly around 0. They both experienced high volatility at times of oil price shocks and market uncertainty. Particularly in 2020 (COVID-19 Pandemic) and 2022/23 (Ukraine Invasion). Although most log return values are close to the mean of ~0 (above or below), there are several outliers in both. It does seem that HP’s log returns are slightly more widely disbursed than PDS. The series both seem to be stationary, since their mean, variance and other characteristics remain the same throughout the entire 5 years.

# plot HP log returns
plot(index(HP_prices)[-1], HP_log_returns, type = "p", col = "orangered", pch = 16, main = "Log Returns of HP vs. Time", xlab = "Date", ylab = "Log Return")

# plot PDS log returns
plot(index(PDS_prices)[-1], PDS_log_returns, type = "p", col = "mediumturquoise", pch = 16, main = "Log Returns of PDS vs. Time", xlab = "Date", ylab = "Log Return")

QUESTION 3.

Quantile-Quantile Plots of Log-Returns vs. Normal Distribution

Both of these normal Q-Q plots do a somewhat decent job at showing that the normal distribution could potentially be used to model the underlying log returns of the stock prices. I don’t believe it’s a very good fit, however, since both of the plots have significant outliers in both the right hand and left hand side tails. Clearly, there is a lot of behaviour in the data pertaining to log returns of the stocks that is not captured or explained by a normal distribution. Although both plots show that most data points’ quantiles match up with those of the normal distribution, the outliers on both sides likely need a different underlying distribution to accurately model them. It is clear from the plots that HP has a greater number of particularly strong outliers in its log returns than PDS does.

# Q-Q Plot for HP_log_returns
qqnorm(HP_log_returns, xlab = "Normal Quantiles", ylab = "Log Return Quantiles", main = "HP | Normal Q-Q Plot for Log Returns", col = "orangered")
qqline(HP_log_returns, col = "black")

# Q-Q Plot for PDS_log_returns
qqnorm(PDS_log_returns, xlab = "Normal Quantiles", ylab = "Log Return Quantiles", main = "PDS | Normal Q-Q Plot for Log Returns", col = "mediumturquoise")
qqline(PDS_log_returns, col = "black")

QUESTION 4.

Quantile-Quantile Plots of Log-Returns vs Student T-Distribution

The Q-Q plot with the best fit for the data seems to be the following:

HP - 4 DoF PDS - 6 DoF

In both these plots, the apparent ‘slope’ of the majority of the observations matched the reference line most closely. Additionally, many of the outliers on the upper and lower tails of the distribution/quantiles were closest to the reference line at these particular values for k (Degrees of Freedom).

# values to try for DoF of the T-distribution
k_vals <- c(2,3,4,5,6,7,8,9,10,20)

# try all different DoF for the HP q-q plots
for (k in k_vals){
  
  # HP theoretical t-distribution quantiles, with DoF = k
  t_quantiles <- qt(ppoints(HP_n), df = k)  
  qqplot(t_quantiles, HP_log_returns_sorted, col = "orangered", xlab = "T-Distribution Quantiles", ylab = "Log Returns", main = paste("HP | Q-Q Plot vs T-Distribution | DoF =", k))
  qqline(HP_log_returns_sorted, col = "black")
  
}

# try all different DoF for the PDS q-q plots
for (k in k_vals){
  
  # PDS theoretical t-distribution quantiles, with DoF = k
  t_quantiles <- qt(ppoints(PDS_n), df = k)  
  qqplot(t_quantiles, PDS_log_returns_sorted, col = "mediumturquoise", xlab = "T-Distribution Quantiles", ylab = "Log Returns", main = paste("PDS | Q-Q Plot vs T-Distribution | DoF =", k))
  qqline(PDS_log_returns_sorted, col = "black")
  
}

QUESTION 5.

Log-Return Contrasts and Similarities

From the plots displayed above, the sorted log returns of PDS as compared to HP are comparatively much more ‘predictable’ in that they seem to adhere much more to the underlying T-distribution. The outliers in the PDS series are far fewer and smaller in magnitude, with the exception of 1 observation on the very far left of the plot. The HP log return series has distinctively more outliers on the left hand tail, representing a lot of unexpectedly large down moves in the stock price. To the same effect, there are far more strong up moves in the HP stock price as well, represented by the numerous outliers in the right hand portion of the log return Q-Q plot. Overall, the log returns of HP seem more widely dispersed and more prone to large outliers than PDS. This could potentially be due to fundamental company factors that make investors more sensitive to changes in the company’s outlook for better or for worse (e.g. riskier revenue sources, weaker balance sheet, more susceptible to macro factors, etc.)

# best fit q-q plots for the log returns vs t-distribution

# HP theoretical t-distribution quantiles, with DoF = 4
t_quantiles <- qt(ppoints(HP_n), df = 4)  
qqplot(t_quantiles, HP_log_returns_sorted, col = "orangered", xlab = "T-Distribution Quantiles", ylab = "Log Returns", main = "HP | Q-Q Plot vs T-Distribution | DoF = 4")
qqline(HP_log_returns_sorted, col = "black")

# PDS theoretical t-distribution quantiles, with DoF = 6
t_quantiles <- qt(ppoints(PDS_n), df = 6)  
qqplot(t_quantiles, PDS_log_returns_sorted, col = "mediumturquoise", xlab = "T-Distribution Quantiles", ylab = "Log Returns", main = "PDS | Q-Q Plot vs T-Distribution | DoF = 6")
qqline(PDS_log_returns_sorted, col = "black")